Finite Group Databasebeta

← #36322136

Subgroups — #36322136

Order 21376.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923178C211yesmaximal, characteristic
2#24923036C211yesmaximal, characteristic, complemented
3#323×C167C211yesmaximal, characteristic, complemented
4C22×D668C411yescharacteristic, complemented
5C22×C22⋊D167—21no—
6C2×C22⋊Dic167—21no—
7C2×(C2×D167)⋊C4—21no—
8C167⋊#95C411yescharacteristic, complemented
9C2×C22⋊C4×C167C2211yescharacteristic
10C22⋊C8×C167—21no—
11C23×D167—21no—
12C2×D668—41no—
13C2×D668—42no—
14C2×C22⋊D167—42no—
15C2×C22⋊D167—42no—
16C2×C22⋊D167—41no—
17C2×C22⋊D167—41no—
18C22×Dic167—21no—
19C22×Dic167—41no—
20C22⋊Dic167—22no—
21C22⋊Dic167—41no—
22(C2×D167)⋊C4—42no—
23C24×C167D411yescharacteristic
24C22×C4×C167C2×C411yescharacteristic
25C22×C4×C167D411yescharacteristic
26C2×C8×C167—41no—
27C22⋊C4×C167—41no—
28C22×D167—22no—
29C22×D167—41no—
30C22×D167—41no—
31C22×D167—41no—
32C22×D167—41no—
33C22×D167—41no—
34C22×D167—41no—
35D668—42no—
36D668—42no—
37C22⋊D167—44no—
38C22⋊D167—44no—
39C2×Dic167—22no—
40C2×Dic167—41no—
41C2×Dic167—41no—
42C2×Dic167—42no—
43C2×Dic167—81no—
44C2×Dic167—81no—
45C23×C167—21no—
46C23×C167—21no—
47C23×C167—41no—
48C23×C167—41no—
49C23×C167C22⋊C411yescharacteristic
50C23×C167D811yescharacteristic
51C23×C167SD1611yescharacteristic
52C2×C4×C167—21no—
53C2×C4×C167—21no—
54C2×C4×C167—41no—
55C2×C4×C167—41no—
56C8×C167—81no—
57C22×C167—21no—
58C22×C167—21no—
59C22×C167—21no—
60C22×C167—21no—
61C22×C167—41no—
62C22×C167—41no—
63C22×C167—41no—
64C22×C167—41no—
65C22×C167—81no—
66C22×C167#9911yescharacteristic
67C22×C167#10211yescharacteristic
68C22×C167#10411yescharacteristic
69C2×D167—42no—
70C2×D167—42no—
71C2×D167—42no—
72C2×D167—81no—
73C2×D167—81no—
74C2×D167—81no—
75C2×D167—81no—
76C4×C167—41no—
77C4×C167—41no—
78Dic167—42no—
79Dic167—82no—
80C2×C167—21no—
81C2×C167—21no—
82C2×C167—41no—
83C2×C167—41no—
84C2×C167#32711yescharacteristic
85C2×C167AΣL(1,𝔽16)11yescharacteristic
86C2×C167#35311yescharacteristic
87D167—82no—
88C167#134311yescharacteristic, complemented, Sylow
89#1343—1671nomaximal, Sylow
90#392—1671no—
91#342—1671no—
92#323D16711yescharacteristic, complemented
93C22×D4—3341no—
94C22×D4—1671no—
95C2×C22⋊C4—3341no—
96C2×C22⋊C4—3341no—
97C2×C22⋊C4C2×D16711yescharacteristic
98C22⋊C8—21no—
99#95—1671no—
100C24—3341no—
101C24D66811yescharacteristic
102C2×D4—6681no—
103C2×D4—6682no—
104C2×D4—6682no—
105C2×D4—6682no—
106C2×D4—6681no—
107C2×D4—6681no—
108C22×C4—3341no—
109C22×C4—6681no—
110C22×C4C4×D16711yescharacteristic
111C22×C4C22⋊D16711yescharacteristic
112C2×C8—41no—
113C22⋊C4—41no—
114C22⋊C4—3342no—
115C22⋊C4—6682no—
116C22⋊C4—6681no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—3342no—
122C23—6681no—
123C23—6681no—
124C23—6681no—
125C23—6681no—
126C23—6681no—
127C23—6681no—
128C23(C2×D167)⋊C411yescharacteristic
129C23D133611yescharacteristic
130C23#1458193911yescharacteristic
131D4—6682no—
132D4—6682no—
133D4—6684no—
134D4—6684no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—13361no—
140C2×C4—13361no—
141C2×C4—3342no—
142C2×C4—6682no—
143C2×C4—6681no—
144C2×C4—6681no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—13361no—
156C22—13361no—
157C22—13361no—
158C22—13361no—
159C22—6682no—
160C22—6682no—
161C22—6682no—
162C22#1912874611yescharacteristic
163C22#1912875411yescharacteristic
164C22#1912875811yescharacteristic
165C4—41no—
166C4—41no—
167C4—13362no—
168C4—6682no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—13362no—
174C2#2492300411yescharacteristic
175C2#2492305711yescharacteristic
176C2#2492306111yescharacteristic
subgroupG/H
centerC22#19128746
commutatorC22×C4×C167C2×C4
FrattiniC2×C22⋊C4C2×D167
Fitting#323×C167C2
socleC22×C167#99
prime radical#24923178C2
cosocleC2×C22⋊C4×C167C22
hypercenter#323D167
hypocenterC167#1343
LCSG ▷ C22×C4×C167 ▷ C23×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #24923036C2semidirect
3. #323×C167C2semidirect
4. C22×D668C4semidirect
8. C167⋊#95C4semidirect
88. C167#1343semidirect
92. #323D167semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#249231782
#249230362
#323×C1672
#1343167